Perturbation To Unified Symmetry And Adiabatic Invariants For Constrained Mechanical Systems

Symmetry is an important and universal nature in mathematics, mechanics and physics. The tiny changes in symmetries, so-called perturbation to symmetries, and adiabatic invariants induced by them play an important role in the research on the quasi-integrability of mechanical systems. Meanwhile, they have wide application in many practical mechanical problems. In the past, the studies on the perturbation to symmetries and adiabatic invariants were limited to these three single symmetrical perturbations: Noether, Lie and Mei symmetrical perturbation, the adiabatic invariants obtained almost belong to the types of Noether and Hojman, and the adiabatic invariants of Mei type have rarely been studied. In this paper, coupled with the development of analytical mechanics, we further study the perturbation to unified symmetry and adiabatic invariants of constrained mechanical systems based on the corresponding theories of symmetries and exact invariants. Firstly, considering d% dt including infinitesimal parameterεobviously after the systems being perturbed, and integrating three single symmetrical perturbations to promote the research level of symmetrical perturbation, we study the perturbation to unified symmetry and adiabatic invariants for continuous constrained mechanical systems. The criteria of the perturbation to unified symmetry for mechanical systems in configuration space, general holonomic, nonholonomic mechanical systems and relativistic Hamilton systems in phase space, and Birkhoffian systems are established. The conditions for existence and the form of Noether adiabatic invariants, generalized Hojman adiabatic invariants (or Hojman adiabatic invariants) and Mei adiabatic invariants, led by the perturbation to unified symmetry of corresponding systems simultaneously, are given. Then, we generalize the theory of the perturbation to symmetries and adiabatic invariants to discrete mechanical systems, propose the concept of discrete higher-order adiabatic invariants, and study the perturbation to unified symmetry and adiabatic invariants for general discrete holonomic systems. The criterion of the perturbation to unified symmetry of the systems is established. The condition for existence and the form of discrete Noether adiabatic invariants, led by the perturbation to unified symmetry of the systems, are given. Finally, the researches in this paper are summarized and theoretical research on symmetries for constrained mechanical systems is prospected.